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George,

The question that I asked in my post may not be THE "question about clearing such impossible triangles" but grant me at least that it is A question!

I paraphrase it here:

Why is it that the application of spherical trigonometric identities to an impossible triangle (one that could not be constructed, even in principle, on the sphere) cannot be relied upon to show inconsistencies in the course of calculation (a sin or cos greater than 1, or a square root of -1)?

The answer I provided was that the lunar distance clearing in this case turns out to be equivalent to operations on a real triangle that can be constructed on the sphere (even if, as noted, this triangle is not one that would arise in clearing lunar distances in practice). A corollary is that neither Maskelyne nor other practitioners can expect the mathematics to alert them to the use of inconsistent starting values.

The triangle I describe has sides of 103, 109 and 161 degrees in length, which does satisfy the condition that the sum of any two sides is greater than the third, yet you seem to suggest that this does not represent a real triangle and hence my argument is implausible. Can you elaborate on what condition prevents this triangle from being constructed?

Regards,
Robin Stuart

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